Mixed Strategy Nash Equilibrium
In Game Theory, the best choice often depends on what others do. Sometimes one choice is clearly the best. Other times, it is not so simple.
- Pure Strategy: we make the same choice every single time.
- Mixed Strategy: we make different choice using probabilities.
This page looks at Mixed Strategy Nash Equilibrium where, players use randomness on purpose. They do this so they aren't easy to predict.
Let's look at a game we all know.
Rock-Paper-Scissors
Let's you and I play rock-paper-scissors! There are three choices:
- Rock beats scissors. Scissors beats paper. Paper beats rock.
- If we both pick the same, we tie.
No single choice will win all the time.
So, how do you play to win?
If you pick rock too often, I will notice this and start picking paper. Then, you lose!
The solution is a Mixed Strategy
Don't pick the same move every time. Pick rock, paper, or scissors randomly.
This keeps you a mystery. I cannot find a pattern to beat you.
Usually, you want to pick each about one-third of the time but people are a little predictable!
- New players often start with Rock (35%)
- Experienced players often start with Paper (35%)
- Scissors is least likely (30%)
Nash Equilibrium
A Nash Equilibrium is when no player is better off by changing only their own strategy.
John Nash came up with this. It proves there is always a "balance" point, even when there isn't one perfect move.
Nash Equilibrium in Mixed Strategies
We can apply Nash Equilibrium to Mixed Strategies!
The goal is play with probabilities so each option gives the same expected result.
When this balance is reached, switching strategies won't improve the outcome. Any predictable change could be used by the opponent.
We call this balance a Mixed Strategy Equilibrium.
Example: Soccer Penalty Kick
Think about a penalty kick:
- The kicker can shoot left, center, or right.
- The goalie can dive left, stay center, or dive right.
The kicker wants a goal. The goalie wants a save.
The best plan? Mix it up randomly. This stops the other person from guessing where you will go.
Mixed strategies stop you from falling into traps. Being random is your shield!
Worked Example: The Tennis Serve
In tennis, a server can serve Wide or down the T. The receiver must guess which way to lean. If the receiver guesses right, the server is less likely to win the point.
To find the Mixed Strategy Nash Equilibrium, the Server must choose a probability that makes the Receiver "indifferent" (meaning the Receiver gets the same result no matter what they choose).
Step 1: Assign Probabilities
- Let p be the probability the Server serves Wide.
- Therefore, 1 − p is the probability the Server serves T.
Step 2: Set the Receiver's Expected Payoffs to Equal
We look at the Receiver's win percentages (the second numbers in the table):
- If Receiver Leans Wide: 70p + 10(1 − p)
- If Receiver Leans T: 20p + 80(1 − p)
For equilibrium, we set these as equal:
70p + 10 − 10p = 20p + 80 − 80p
Step 3: Solve for p
- Simplify: 60p + 10 = 80 - 60p
- Add 60p to both sides: 120p + 10 = 80
- Subtract 10: 120p = 70
- Divide: p = 70/120 or 0.58
The Result: To be unpredictable and unexploitable, the Server should serve Wide 58% of the time and serve T 42% of the time.
If the Server uses this exact mix, the Receiver will win 45% of points regardless of whether they lean Wide or T. The Server has successfully protected their strategy!
Mastering Game Theory
The big idea is easy: when no single choice is perfect, randomness can be the best strategy.
By using mixed strategies we aren't guessing: we are strategically choosing when to be unpredictable.
This powerful idea helps explain behavior in games, markets, sports, and real-world decision making.